Fixed-Point Definability and Polynomial Time on Chordal Graphs and Line Graphs

TL;DR

This paper investigates the limits of fixed-point logic with counting in capturing polynomial time on specific graph classes, showing it fails on chordal and line graphs individually but succeeds on their intersection.

Contribution

It proves that fixed-point logic with counting cannot capture polynomial time on chordal or line graphs alone, but can on graphs that are both chordal and line graphs.

Findings

Fixed-point logic with counting does not capture polynomial time on chordal graphs.

Fixed-point logic with counting does not capture polynomial time on line graphs.

It captures polynomial time on graphs that are both chordal and line graphs.

Abstract

The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988. It is still wide open and regarded as one of the main open problems in finite model theory and database theory. Partial results have been obtained for specific classes of structures. In particular, it is known that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. The introductory part of this paper is a short survey of the state-of-the-art in the quest for a logic capturing polynomial time. The main part of the paper is concerned with classes of graphs defined by excluding induced subgraphs. Two of the most fundamental such classes are the class of chordal graphs and the class of line graphs. We prove that capturing polynomial time on either of these classes is as hard as capturing it on the class of all graphs. In particular, this implies that fixed-point logic with counting does not capture polynomial time on these classes. Then we prove that fixed-point logic with counting does capture polynomial time on the class of all graphs that are both chordal and line graphs.